Why can't the growth rate be higher than the discount rate?

Why can't the discount rate be lower than the growth rate in terminal value?

What is the theoretical reason for it.

Thanks.


Ways to Calculate Terminal Value

Terminal value is an important part in determining company valuation. Before digging in to the theoretical explanation to the above question, here's a quick review of the calculation. Depending on various factors, you may want to use an exit multiple or perpetual growth method, such as the Gordon Growth Model for determining terminal value in a DCF model.

  • Perpetual Growth: Use when company is in its long-term, mature growth phase
    • Terminal Value = Last Year Free Cash Flow x ((1 + Terminal Growth Rate) / (WACC - Terminal Growth Rate))
  • Exit Multiple: Use when company is not yet in steady growth phase or when market has a good idea of acquisition value (ex: LBO)

For more information on how to find your growth rate and discount rate, check out these posts:

Remember, no matter what formula and inputs you use, it is just an approximation or attempt to model a complex real world process.

How Growth Rate and Discount Rate Impact Terminal Value Formula

From a simple mathematical perspective, the growth rate can't be higher than the discount rate because it would give you a negative terminal value. From a theoretical perspective, Certified Investment Banking Professional – 1st Year Associate jhoratio explains:

Growth rates can exceed the cost of capital for very short periods of time, but we're talking about a growth rate IN PERPETUITY here. Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it.

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Comments (33)

 
Mar 15, 2010 - 4:45pm

The model does not work because it would give you a negative number (impossible). You would need to use EBITDA and an exit multiple to find terminal value.

"Greed, in all of its forms; greed for life, for money, for love, for knowledge has marked the upward surge of mankind. And greed, you mark my words, will not only save Teldar Paper, but that other malfunctioning corporation called the USA."
  • 1
 
Mar 15, 2010 - 4:47pm

Uh, it has to be, otherwise you get negative terminal value.

n - terminal year, r - discount rate, g - growth rate

Perpetuity PV = (FCF @ n+1) / (r - g)

If r

 
Mar 15, 2010 - 4:54pm

Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

 
Mar 15, 2010 - 5:09pm

nick_123:
Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

A company's long-term growth rate isn't going to surpass the amount required for investors of all securities to take on the risk. Just think about it, first off it's a projection based on assumptions, so it's not a reflection of actual returns or growth.

 
Mar 15, 2010 - 7:48pm

BobbyLight:
nick_123:
Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

A company's long-term growth rate isn't going to surpass the amount required for investors of all securities to take on the risk. Just think about it, first off it's a projection based on assumptions, so it's not a reflection of actual returns or growth.

Hmmm, I'm still not seeing it. Why wouldn't a company's growth rate be able to surpass the required return of investors?

After all, growth and risk are driven by different variables.

 
Mar 15, 2010 - 8:31pm

If a company were to grow faster than the expected rate of return in perpetuity, in effect growing faster than the market itself, then the company would be on pace to eventually become larger than the entire market. Impossible.

 
Best Response
Mar 15, 2010 - 8:35pm

Nick, growth rates can exceed the cost of capital for very short periods of time, but we're talking about a growth rate IN PERPETUITY here. This is kind of like asking, "why don't trees just keep growing past the clouds?" or "why can't a stock be worth less than zero?" You're just playing with numbers here and have forgotten the underlying reality. Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it. Wow! Talk about a conglomerate! Seriously, unless you think this is a likely scenario, it just simply cannot be that the growth rate exceeds the risk. Remember all these formulas are just mathematical APPROXIMATIONS of incredibly complex real world processes. Don't let the tail wag the dog.

 
Feb 25, 2017 - 9:27am

With an answer like that, you should be working at a hedge fund.

Absolute truths don't exist... celebrated opinions do.
  • 1
 
Mar 1, 2017 - 1:48am

jhoratio:

Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it. Wow! Talk about a conglomerate! ... Remember all these formulas are just mathematical APPROXIMATIONS of incredibly complex real world processes. Don't let the tail wag the dog.

Beautiful. Great answer.

Maximum effort.
  • 4
 
Jan 13, 2021 - 2:15pm

A company with 300% annual growth rate, like Intuit had at the start, for example, shouldn't have a discount rate of 301%. It don't work for startups. 

Is there a dcf based valuation method for startups or intrapreneurial projects within an established firm?

 
Jan 13, 2021 - 3:07pm

+1 SB. Thank you posting a good part of the answer.

....this whole thread is really really sad. It shows that a lot of people know modeling well but don't understand the underlying...here go some additional details:

1. Economic reasons for this include the attraction of competition. If you're growing at a tremendous rate, it's not that investors just want to invest in your company. It's that competitors will jump into your industry as well.

2. There is a limited amount of market share out there. If you're growing at 20% per year, your model will soon have more customers than exist for the product on the entire planet.

3. Reasons one and two lead to a predictable life cycle of a company. We have seen this over and over again with companies that initially have super strong growth rates and as they become bigger and bigger set into a fairly predictable steady state. Consumer goods are a great way to think about this. At one point, Coca-Cola was growing like crazy. As it became a mature and larger company, it's simply not going to sell 20% cokes next year. The opportunity set and markets have been for the large part exhausted. Or think of even something like a Microsoft. It grew like crazy and now it's a blue chip. Low terminal growth rate is the fate of us all if we do well enough.

4. Lastly, as someone else said later on here, when you've completely exhausted your obvious opportunities and taken all the share you can, guess how you grow? Some relationship to GDP which will almost necessarily be below your discount rate.

 
Mar 15, 2010 - 8:55pm

Perpetuity Growth rate higher than required rate of return = a bubble that will never burst.. like puff in the microwave...

Signs of Recession: Banker: “Where’s me Bonus?” Yuppie: “Whadya mean I have to actually work?” Fox Rock Mum: “Lidl’s the place to be seen in now!” Cowen: “It’s not my fault that me and my party are complete f**k-ups - it’s the recession silly!”
 
Mar 16, 2010 - 12:18am

If you have any perpetual yearly cash flow that grows at a rate greater than the discount rate, your NPV will be infinite. Think about it this way - every future year's present value will be greater than the previous - because your cash flow is growing faster than you can discount it - and thus you'll will not obtain a finite net present value.

 
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Oct 3, 2015 - 5:12pm

Question:

i have a project planned to start on 1 jan 2010, is able to produce revenue cash flow starting from 1million on 1 jan 2011 and this is expected to grow at 13% per year until 1 jan 2020.

The initial cost of the project is 20million. Beta: 1.1, risk free rate: 1.7%, risk premium is 9.5%.

So the cost of capital should be 0.017 + 1.1(0.095) = 0.1215

but it is lower than the growth rate of 13%.

is my calculation wrong?

 
Mar 1, 2017 - 4:05am

This is a purely mathematical question, it can be answered with zero economic reasoning (although you can go further with it to give the economic interpretation like the guys went here). The sum of perpetual "cash flows" (or whatever) growing at a constant rate and being discounted through time only converges to a value (the "V = F/(k-g)" formula) if the growth rate is lower than the discount rate. If it is higher, this value (the sum of the cash flows, not the formula, which is conditional to this specific case: k > g) is infinite (as all the terms of the sum are higher than one, and the sum is infinite). It can be negative if the numerator is negative (the "cash flows"), and if the growth rate is higher than the discount rate, then it is infinitely negative.

"Never believe in anything until it has been officially denied"
  • 2
 
Jun 16, 2018 - 10:31am

absolutely, imo you can model any scenario you like and can defend. however these assumptions (as always) need to be reasonable, meaning that the stages should not have abrupt changes in the pace of growth. A way to prevent this is adding a mechanical appendix to the initial forecast (say 6 years initial forecast + 4 years appendix), where growth rates are slowly converging to the perpetuity growth rate assumed.

i believe this method often increases the validity of a model

 
Jun 16, 2018 - 11:38pm

This response is from a purely conceptual economic point of view (ignoring all of the other valid practical arguments like trees don't grow to the sky and mathematical arguments around divergent series).

If you think about a discount rate as a required rate of return, this becomes an easier question to understand.

Roughly speaking, a security's return / discount rate =
1. yield plus
2. operations growth (EBITDA, FCF, whatever) plus
3. changes in multiple.

Re #3, in perpetuity, you don't get any of the effects of changes in multiple. So we just have yield + operations growth.

Re #1, if we're talking about a strictly positive FCF business, the yield has to be positive.

Therefore, your growth rate must be less than your overall rate of return since your rate of return should take into account both growth and positive yield.

 
Jun 16, 2018 - 11:45pm

The standard perpetuity with growth formula no longer is valid. It does not give you a negative value and anyone who says so is mathematically mistaken. The formula is derived from a geometric series, with geometric part less than 1. If r

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